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Difference between revisions of "2018 USAJMO Problems/Problem 3"

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Now, <math>DP</math> is the radical axis of the circumcircles of <math>\triangle EDP</math> and <math>\triangle FDP.</math> Since <math>B</math> lies on <math>DP,</math> and <math>E, Q</math> lie on the circumcircle of <math>\triangle EPD</math> and <math>F, R</math> lie on the circumcircle of <math>\triangle FPD,</math> we have that <cmath>BF \cdot BR = BE \cdot BQ.</cmath> However, <math>BF=BE,</math> so <math>BR=BQ.</math> Since <math>E, B, Q</math> are collinear and so are <math>F, B, R</math> we can add these <math>2</math> equations to get <cmath>EQ=BE+BQ=BF+BR=FR,</cmath> which completes the proof.
 
Now, <math>DP</math> is the radical axis of the circumcircles of <math>\triangle EDP</math> and <math>\triangle FDP.</math> Since <math>B</math> lies on <math>DP,</math> and <math>E, Q</math> lie on the circumcircle of <math>\triangle EPD</math> and <math>F, R</math> lie on the circumcircle of <math>\triangle FPD,</math> we have that <cmath>BF \cdot BR = BE \cdot BQ.</cmath> However, <math>BF=BE,</math> so <math>BR=BQ.</math> Since <math>E, B, Q</math> are collinear and so are <math>F, B, R</math> we can add these <math>2</math> equations to get <cmath>EQ=BE+BQ=BF+BR=FR,</cmath> which completes the proof.
  
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~nukelauncher
 +
(Monday G. Fern)
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 +
==Solution 2==
 +
We begin with the following claims.
 +
 +
Claim. <math>B</math> is the circumcenter of <math>\triangle DEF</math>.
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Proof. By reflection <math>BD=BE=BF</math>.
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 +
Claim. <math>A</math> is the circumcenter of <math>\triangle EPD</math>.
 +
Proof. First, we have
 +
    <cmath>\angle DPE = 180^{\circ}-\angle PDE - \angle PED = 180^{\circ}-\angle BED - \frac{\angle DBF}{2} = \angle 180^{\circ}-\angle BED - \frac{180^{\circ}-2\angle BFD}{2} = 90^{\circ}-\angle BDE + \angle DFB</cmath>
 +
 +
Then
 +
 +
    <cmath> \angle ADE = \angle ADP - \angle EDP = \angle ADB - \angle EDB = \angle ACB - \angle BDE = \angle FDB - \angle BDE = \angle DFB - \angle DEB = \angle DPE - 90^{\circ} = \frac{180^{\circ}-2(180^{\circ}-\angle DPE)}{2}</cmath>
 +
Then <math>2\angle ADE = 180^{\circ}-2(180^{\circ}-\angle DPE) \implies \angle DAE = 2(180^{\circ}-\angle DPE)</math>. This is enough to imply what we desire. \newline
 +
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Claim. <math>C</math> is the circumcenter of <math>\triangle FPD</math>.
 +
Proof. Similar to above.
 +
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Claim. <math>E, B, Q</math> are collinear.
 +
Proof. We have
 +
    <cmath>\angle ABE = \angle ABD = \angle AQD = \angle ADQ = 180^{\circ}-\angle ABQ</cmath>
 +
 +
Claim.<math>F, B, R</math> are collinear.
 +
Proof. Similar to above.
 +
 +
Since <math>DEPQ</math> is cyclic, <math>\triangle EBP \sim \triangle DBQ</math>. However, <math>BE=BD</math>, so <math>BP=BQ</math>. Similarly, <math>BP=BR</math>. Finishing, we have <cmath>EQ = EB + BQ = BD + BP = FB + BR = FR</cmath>, as desired. <math>\blacksquare</math> ~MSC
  
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 19:30, 4 November 2020

Problem

($*$) Let $ABCD$ be a quadrilateral inscribed in circle $\omega$ with $\overline{AC} \perp \overline{BD}$. Let $E$ and $F$ be the reflections of $D$ over lines $BA$ and $BC$, respectively, and let $P$ be the intersection of lines $BD$ and $EF$. Suppose that the circumcircle of $\triangle EPD$ meets $\omega$ at $D$ and $Q$, and the circumcircle of $\triangle FPD$ meets $\omega$ at $D$ and $R$. Show that $EQ = FR$.

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Solution 1

First we have that $BE=BD=BF$ by the definition of a reflection. Let $\angle DEB = \alpha$ and $\angle DFB = \beta.$ Since $\triangle DBE$ is isosceles we have $\angle BDE = \alpha.$ Also, we see that $\angle BDE = \angle CAB = \angle CDB = \alpha,$ using similar triangles and the property of cyclic quadrilaterals. Similarly, \[\angle DFB = \angle FDB = \angle ACB = \angle ADB = \beta.\] Now, from $BE=BD=BF$ we know that $B$ is the circumcenter of $\triangle DEF.$ Using the properties of the circumcenter and some elementary angle chasing, we find that \[\angle DPE = 90^{\circ} + \beta - \alpha.\]

Now, we claim that $Q$ is the intersection of ray $\overrightarrow{EB}$ and the circumcircle of $ABCD.$ To prove this, we just need to show that $DEPQ$ is cyclic by this definition of $Q.$ We have that \[\angle DQE = \angle DCB = \angle DCA + \angle ACB = (90^{\circ}-\alpha)+\beta.\] We also have from before that \[\angle DPE = 90+\beta-\alpha,\] so $\angle DQE=\angle DPE$ and this proves the claim.

We can use a similar proof to show that $F, B, R$ are collinear.

Now, $DP$ is the radical axis of the circumcircles of $\triangle EDP$ and $\triangle FDP.$ Since $B$ lies on $DP,$ and $E, Q$ lie on the circumcircle of $\triangle EPD$ and $F, R$ lie on the circumcircle of $\triangle FPD,$ we have that \[BF \cdot BR = BE \cdot BQ.\] However, $BF=BE,$ so $BR=BQ.$ Since $E, B, Q$ are collinear and so are $F, B, R$ we can add these $2$ equations to get \[EQ=BE+BQ=BF+BR=FR,\] which completes the proof.

~nukelauncher (Monday G. Fern)

Solution 2

We begin with the following claims.

Claim. $B$ is the circumcenter of $\triangle DEF$. Proof. By reflection $BD=BE=BF$.

Claim. $A$ is the circumcenter of $\triangle EPD$. Proof. First, we have 

   \[\angle DPE = 180^{\circ}-\angle PDE - \angle PED = 180^{\circ}-\angle BED - \frac{\angle DBF}{2} = \angle 180^{\circ}-\angle BED - \frac{180^{\circ}-2\angle BFD}{2} = 90^{\circ}-\angle BDE + \angle DFB\]

Then 

   \[\angle ADE = \angle ADP - \angle EDP = \angle ADB - \angle EDB = \angle ACB - \angle BDE = \angle FDB - \angle BDE = \angle DFB - \angle DEB = \angle DPE - 90^{\circ} = \frac{180^{\circ}-2(180^{\circ}-\angle DPE)}{2}\]

Then $2\angle ADE = 180^{\circ}-2(180^{\circ}-\angle DPE) \implies \angle DAE = 2(180^{\circ}-\angle DPE)$. This is enough to imply what we desire. \newline

Claim. $C$ is the circumcenter of $\triangle FPD$. Proof. Similar to above.

Claim. $E, B, Q$ are collinear. Proof. We have 

   \[\angle ABE = \angle ABD = \angle AQD = \angle ADQ = 180^{\circ}-\angle ABQ\]

Claim.$F, B, R$ are collinear. Proof. Similar to above.

Since $DEPQ$ is cyclic, $\triangle EBP \sim \triangle DBQ$. However, $BE=BD$, so $BP=BQ$. Similarly, $BP=BR$. Finishing, we have \[EQ = EB + BQ = BD + BP = FB + BR = FR\], as desired. $\blacksquare$ ~MSC

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

See also

2018 USAJMO (Problems • Resources)
Preceded by
Problem 2 		Followed by
Problem 4
1 • 2 • 3 • 4 • 5 • 6
All USAJMO Problems and Solutions
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